0
votes
1answer
13 views

A criterion for complete lattice.

Is there an infinite partially ordered set $(X,\le)$, in which for each $A\subseteq X$, either $\inf A$ or $\sup A$ exists but for some $A\subseteq X$ either $\inf A$ or $\sup A$ does not exist.
0
votes
1answer
73 views

Is this poset a lattice?

Given the set $\Bbb Z^+\times\Bbb Z^+$ and the relation $$\begin{align*} (x_1,x_2)\,R\,(y_1,y_2)\iff &(x_1+x_2 < y_1 + y_2)\\ &\text{ OR }(x_1 + x_2 = y_1 + y_2\text{ AND }x_1 \le y_1)\;: ...
2
votes
1answer
136 views

Smallest Congruence Relation generated by a set

$\newcommand{\cl}{\operatorname{cl}}$ Let $R \subset S \times S$ be a binary relation, the smallest i) reflexive relation containing it is $$ \cl_\mathrm{ref} = R \cup \{ (x,x) : x \in S \} $$ ii) ...
0
votes
1answer
61 views

an infinite queue preserving equality.

Is there any well-ordered set $(A,\leq)$ such that: $(A,\leq^{-1})$ is well-ordered. $A$ is infinite. there's exactly one function $\theta:A\rightarrow \{0,1\}$ such that 1) for each $a < M$, ...
2
votes
2answers
730 views

Problem about Hasse diagrams

Can someone help me to solve this problem. Are these Hasse diagrams lattices?
0
votes
1answer
57 views

Confusion with the lattice formed by a partition

I was referring to this article here related to the formation of a complete lattice by the partitions of a set. The article has stated that the partitions not only form the lattice for themselves but ...
0
votes
2answers
192 views

Confusion about the lattice formed by an equivalence relation

I am a beginner in this field. Actually, I am studying about equivalence relation. I found that the set of all equivalence relations possible on set A form a relation. If R1 and R2 are two ...
4
votes
2answers
193 views

An analogy between subgroups and equivalence relations.

I have noticed a certain analogy between subgroups of a group $G$ and equivalence relations on a set $X$. I would like to know if there's an explanation for this analogy or a common generalization of ...
2
votes
1answer
141 views

Equivalence relation independence from G.C.Rota paper?

On page 4 of Many Lives of Lattice Theory the author wrote: "Two equivalence relations on a set are said to be independent when every equivalence class of the first meets every equivalence class of ...